♨️Thermodynamics of Fluids Unit 8 – Phase Equilibria and Stability

Key Concepts and Definitions

• Phase equilibrium occurs when two or more phases coexist in a system with no net transfer of mass or energy between them
• Stability refers to a system's ability to maintain its state when subjected to small perturbations
• Gibbs free energy ($G$) is a thermodynamic potential that determines the stability and spontaneity of a process at constant temperature and pressure
  • Defined as $G = H - TS$, where $H$ is enthalpy, $T$ is temperature, and $S$ is entropy
• Chemical potential ($\mu$) is the partial molar Gibbs free energy and represents the change in $G$ when a component is added to a system
• Fugacity ($f$) is a measure of a component's tendency to escape from a phase, related to its chemical potential
• Activity ($a$) is the effective concentration of a component in a mixture, accounting for non-ideal behavior
• Raoult's law states that the vapor pressure of a component in an ideal solution is proportional to its mole fraction in the liquid phase
• Henry's law describes the solubility of a gas in a liquid at low concentrations, where the gas's partial pressure is proportional to its mole fraction in the liquid

Thermodynamic Principles

• First law of thermodynamics states that energy cannot be created or destroyed, only converted from one form to another
  • Expressed as $\Delta U = Q + W$, where $\Delta U$ is the change in internal energy, $Q$ is heat added, and $W$ is work done
• Second law of thermodynamics introduces the concept of entropy ($S$), a measure of the system's disorder or randomness
  • States that the entropy of an isolated system always increases or remains constant
• Third law of thermodynamics establishes the absolute zero of entropy, stating that a perfect crystal at 0 K has zero entropy
• Fundamental equation of thermodynamics relates the change in internal energy to changes in entropy, volume, and composition
  • $dU = TdS - PdV + \sum \mu_i dN_i$, where $P$ is pressure, $V$ is volume, and $N_i$ is the number of moles of component $i$
• Maxwell relations are derived from the fundamental equation and relate partial derivatives of thermodynamic properties
• Gibbs-Duhem equation constrains the changes in chemical potentials of components in a mixture at constant temperature and pressure

Phase Diagrams and Their Interpretation

• Phase diagrams graphically represent the equilibrium states of a system as a function of thermodynamic variables (e.g., temperature, pressure, composition)
• Common types include pressure-temperature (P-T), temperature-composition (T-x), and pressure-composition (P-x) diagrams
• Phase boundaries separate regions of the diagram where different phases are stable
  • Represent conditions at which two or more phases coexist in equilibrium
• Triple point is a unique condition where three phases coexist in equilibrium
• Critical point represents the highest temperature and pressure at which vapor-liquid equilibrium can exist
• Tie lines connect the compositions of coexisting phases in a two-phase region
• Lever rule determines the relative amounts of each phase present at a given overall composition
• Eutectic point is the lowest temperature at which a liquid phase can exist in a binary system

Gibbs Phase Rule

• Gibbs phase rule relates the number of degrees of freedom ($F$), components ($C$), and phases ($P$) in a system at equilibrium
  • Expressed as $F = C - P + 2$
• Degrees of freedom represent the number of independent variables that can be changed without altering the number of phases in equilibrium
• For a single-component system, the maximum number of degrees of freedom is 2 (usually temperature and pressure)
• In a binary system, the maximum number of degrees of freedom is 3 (temperature, pressure, and composition)
• Applying the phase rule helps determine the variance of a system and the conditions under which phase transitions occur

Equilibrium Conditions

• Chemical equilibrium is achieved when the chemical potentials of each component are equal in all phases
  • Expressed as $\mu_i^\alpha = \mu_i^\beta = ... = \mu_i^\pi$, where $\alpha$, $\beta$, and $\pi$ represent different phases
• Thermal equilibrium requires the temperature to be uniform throughout the system
• Mechanical equilibrium is reached when the pressure is the same in all phases
• Gibbs energy minimization principle states that a system at constant temperature and pressure will minimize its Gibbs free energy at equilibrium
• Common models for describing phase equilibria include ideal solution, regular solution, and activity coefficient models (e.g., Wilson, NRTL, UNIQUAC)
• Equilibrium constants ($K$) relate the activities or fugacities of components in different phases at equilibrium
  • For vapor-liquid equilibrium, $K_i = y_i / x_i$, where $y_i$ and $x_i$ are the mole fractions of component $i$ in the vapor and liquid phases, respectively

Stability Criteria

• Thermodynamic stability requires that a system's Gibbs free energy is at a global minimum with respect to all possible perturbations
• Mathematical conditions for stability involve the second derivatives of Gibbs free energy with respect to relevant variables
  • For example, $(\partial^2 G / \partial T^2)_P > 0$ and $(\partial^2 G / \partial P^2)_T > 0$ for thermal and mechanical stability, respectively
• Spinodal curve represents the limit of metastability, beyond which a phase becomes unstable and spontaneously separates
• Binodal curve (or coexistence curve) represents the equilibrium compositions of two phases in a binary system
• Metastable regions exist between the binodal and spinodal curves, where a phase is stable with respect to small perturbations but unstable to large fluctuations
• Gibbs stability criteria involve the determinant of the Hessian matrix of second derivatives of Gibbs free energy
  • A phase is stable if the determinant and all its principal minors are positive

Applications in Real Systems

• Vapor-liquid equilibrium (VLE) is crucial in the design and operation of distillation columns, absorbers, and strippers
• Liquid-liquid extraction (LLE) relies on the equilibrium distribution of a solute between two immiscible liquid phases
• Solid-liquid equilibrium (SLE) is important in crystallization processes, such as in the pharmaceutical and food industries
• Gas hydrates are solid compounds that form at high pressures and low temperatures, relevant in natural gas processing and flow assurance
• Adsorption processes involve the equilibrium distribution of a solute between a fluid phase and a solid surface
• Phase behavior of polymer solutions and blends is essential in the plastics and materials industries
• Geochemical systems, such as mineral-fluid equilibria, are important in understanding the formation and evolution of Earth's crust
• Biological systems, such as lipid membranes and protein folding, involve complex phase behavior and stability considerations

Problem-Solving Techniques

• Identify the number of components, phases, and degrees of freedom using Gibbs phase rule
• Determine the appropriate thermodynamic model for the system (e.g., ideal solution, activity coefficient models)
• Write equilibrium conditions based on equality of chemical potentials or fugacities
• Use mass balances and phase diagram information to set up a system of equations
• Solve the system of equations numerically or analytically to obtain equilibrium compositions and phase amounts
• Interpret the results in terms of phase stability, critical points, and other relevant features
• Perform sensitivity analyses to assess the impact of uncertainties in model parameters or input data
• Validate the results against experimental data or known phase behavior trends
• Consider the limitations and assumptions of the chosen thermodynamic model and their implications for the solution